Group homomorphism/Matrix 3 4 1 2/Bijective for Q, not for Z/Exercise

We consider the matrix

$\left(\begin{array}{cc}3& 4\\ 1& 2\end{array}\right).$

Show that this matrix defines a group homomorphism from ${\mathbb{Q}}^2$ to ${\mathbb{Q}}^2$, and from ${\mathbb{Z}}^2$ to ${\mathbb{Z}}^2$ as well. Study this group homomorphism with respect to injectivity and surjectivity.

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